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๏ปฟPianote / Chords / UPDATED Mar 9, 2023 C-sharp major/minor and D-flat major/minor are essentially the same keys using the same pitches but can be named either way. Click on the chord symbol for a diagram and explanation of each chord type Key of C-Sharp Cโฏ Cโฏm Cโฏsus2 Cโฏsus4 Cโฏmaj7 Cโฏm7 Cโฏ7 Cโฏdim7 Cโฏm7โญ5 Key of D-Flat Dโญ Dโญm Dโญsus2 Dโญsus4 Dโญmaj7 Dโญm7 Dโญ7 Dโญdim7 Dโญm7โญ5 Cโฏ MAJOR TRIAD Chord Symbol Cโฏ or Cโฏmaj The Cโฏ major triad consists of a root Cโฏ, third Eโฏ, and fifth Gโฏ. The distance between the root and the third is a major third interval or four half-steps, and the distance between the third and the fifth is a minor third interval or three half-steps. Major triads have a โhappyโ sound. Cโฏ Major Triad Root Position Cโฏ Major Triad 1st Inversion Cโฏ Major Triad 2nd Inversion Dโญ MAJOR TRIAD Chord Symbol Dโญ or Dโญmaj The Dโญ major triad consists of a root Dโญ, third F, and fifth Aโญ. The distance between the root and the third is a major third interval or four half-steps, and the distance between the third and the fifth is a minor third interval or three half-steps. Major triads have a โhappyโ sound. Dโญ Major Triad Root Position Dโญ Major Triad 1st Inversion Dโญ Major Triad 2nd Inversion Cโฏ MINOR TRIAD Chord Symbol Cโฏm The Cโฏ minor triad consists of a root Cโฏ, third E, and fifth Gโฏ. The distance between the root and the third is a minor third interval or three half-steps, and the distance between the third and the fifth is a major third interval or four half-steps. Minor triads have a โsadโ sound. Cโฏ Minor Triad Root Position Cโฏ Minor Triad 1st Inversion Cโฏ Minor Triad 2nd Inversion Dโญ MINOR TRIAD Chord Symbol Dโญm The Dโญ minor triad consists of a root Dโญ, third Fโญ, and fifth Aโญ. The distance between the root and the third is a minor third interval or three half-steps, and the distance between the third and the fifth is a major third interval or four half-steps. Minor triads have a โsadโ sound. Dโญ Minor Triad Root Position Dโญ Minor Triad 1st Inversion Dโญ Minor Triad 2nd Inversion Cโฏ SUSPENDED 2 Chord Symbol Cโฏsus2 In the Cโฏsus2 chord, the third of the Cโฏ major or minor chord Eโฏ or E is replaced โsuspendedโ with the second note Dโฏ of the Cโฏ major scale. Root Position Dโญ SUSPENDED 2 Chord Symbol Dโญsus2 In the Dโญsus2 chord, the third F or Fโญ of the Dโญ major or minor chord is replaced โsuspendedโ with the second note Eโญ of the Dโญ major scale. Root Position Cโฏ SUSPENDED 4 Chord Symbol Cโฏsus4 In the Cโฏsus4 chord, the third of the Cโฏ major or minor chord Eโฏ or E is replaced โsuspendedโ with the fourth note Fโฏ of the Cโฏ major scale. Root Position Dโญ SUSPENDED 4 Chord Symbol Dโญsus2 In the Dโญsus4 chord, the third F or Fโญ of the Dโญ major or minor chord is replaced โsuspendedโ with the fourth note Gโญ of the Dโญ major scale. Root Position Cโฏ MAJOR 7 Chord Symbol Cโฏmaj7 or Cโฏฮ7 A major 7 chord is a major triad with an added seventh. The distance between the root and the seventh is a major 7th interval. Cโฏmaj7 Root Position Cโฏmaj7 1st Inversion Cโฏmaj7 2nd Inversion Cโฏmaj7 3rd Inversion Dโญ MAJOR 7 Chord Symbol Dโญmaj7 or Dโญฮ7 A major 7 chord is a major triad with an added seventh. The distance between the root and the seventh is a major 7th interval. Dโญmaj7 Root Position Dโญmaj7 1st Inversion Dโญmaj7 2nd Inversion Dโญmaj7 3rd Inversion Cโฏ MINOR 7 Chord Symbol Cโฏm7 A minor 7 chord is a minor triad with an added seventh. The distance between the root and the seventh is a minor 7th interval. Cโฏm7 Root Position Cโฏm7 1st Inversion Cโฏm7 2nd Inversion Cโฏm7 3rd Inversion Dโญ MINOR 7 Chord Symbol Dโญm7 A minor 7 chord is a minor triad with an added seventh. The distance between the root and the seventh is a minor 7th interval. Dโญm7 Root Position Dโญm7 1st Inversion Dโญm7 2nd Inversion Dโญm7 3rd Inversion Cโฏ DOMINANT 7TH Chord Symbol Cโฏ7 A dominant 7th chord is a major triad with an added seventh, where the distance between the root and the seventh is a minor 7th interval. You can also think of dominant 7th chords as being built on the fifth note of a major scale and following that scaleโs key signature. For example, Cโฏ7 is built on Cโฏ, the fifth note of F-sharp major, and follows F-sharp majorโs key signature Fโฏ, Cโฏ, Gโฏ, Dโฏ, Aโฏ, Eโฏ. Cโฏ7 Root Position Cโฏ7 1st Inversion Cโฏ7 2nd Inversion Cโฏ7 3rd Inversion Dโญ DOMINANT 7TH Chord Symbol Dโญ7 A dominant 7th chord is a major triad with an added seventh, where the distance between the root and the seventh is a minor 7th interval. You can also think of dominant 7th chords as being built on the fifth note of a major scale and following that scaleโs key signature. For example, Dโญ7 is built on Dโญ, the fifth note of G-flat major, and follows G-flat majorโs key signature Bโญ, Eโญ, Aโญ, Dโญ, Gโญ, Cโญ. Dโญ7 Root Position Dโญ7 1st Inversion Dโญ7 2nd Inversion Dโญ7 3rd Inversion Cโฏ DIMINISHED 7TH Chord Symbol Cโฏdim7 A diminished 7th chord is a four-note-chord where each note is a minor third apart. You can think of diminished 7th chords as a โstack of minor thirds.โ Cโฏdim7 Root Position Cโฏdim7 1st Inversion Cโฏdim7 2nd Inversion Cโฏdim7 3rd Inversion Dโญ DIMINISHED 7TH Chord Symbol Dโญdim7 A diminished 7th chord is a four-note-chord where each note is a minor third apart. You can think of diminished 7th chords as a โstack of minor thirds.โ Dโญdim7 Root Position Dโญdim7 1st Inversion Dโญdim7 2nd Inversion Dโญdim7 3rd Inversion Cโฏ HALF DIMINISHED 7TH Chord Symbol Cโฏm7โญ5 The half-diminished chord is also called the โminor seven flat fiveโ chord. It is a minor 7th chord where the fifth is lowered by a half-step. Cโฏm7โญ5 Root Position Cโฏm7โญ5 1st Inversion Cโฏm7โญ5 2nd Inversion Cโฏm7โญ5 3rd Inversion Dโญ HALF DIMINISHED 7TH Chord Symbol Dโญm7โญ5 The half-diminished chord is also called the โminor seven flat fiveโ chord. It is a minor 7th chord where the fifth is lowered by a half-step. Dโญm7โญ5 Root Position Dโญm7โญ5 1st Inversion Dโญm7โญ5 2nd Inversion Dโญm7โญ5 3rd Inversion ๐น Your Go-To Place for All Things PianoSubscribe to The Note for exclusive interviews, fascinating articles, and inspiring lessons delivered straight to your inbox. Unsubscribe at any time. Pianote is the Ultimate Online Piano Lessons Experienceโข. Learn at your own pace, get expert lessons from real teachers and world-class pianists, and join a community of supportive piano players. Learn more about becoming a Member.
In this explainer, we will learn how to identify the relationship between chords that are equal or different in length and the center of a circle and use the properties of the chords in congruent circles to solve begin by recalling that perpendicular bisectors of chords go through the center of the circle. Let us draw a diagram portraying this the diagram above, the blue line segment perpendicularly bisects chord ๐ด๐ต. We note that this line goes through the center ๐ and, hence, defines the perpendicular distance between the center and the Distance of a Chord from the CenterThe distance of a chord from the center of the circle is measured by the length of the line segment from the center that intersects perpendicularly with the the diagram above, let us label the midpoint of chord ๐ด๐ต, which is where the blue line perpendicularly intersect with the chord. Also, we will add radius โณ๐๐ถ๐ด is a right triangle, we can use the Pythagorean theorem to find length ๐ด๐ถ from radius ๐ด๐ and distance ๐๐ถ. Since ๐ถ is the midpoint of chord ๐ด๐ต, we know that ๐ด๐ต=2๐ด๐ถ. Hence, if we are given the radius of the circle and the distance of a chord from the center of the circle, we can use this method to find the length of the chord. Rather than explicitly writing out this computation, we will focus on the qualitative relationship between the lengths of chords and their distance from the center of the circle in this two different chords in the same circle as in the diagram ๐๐ด and ๐๐ท are radii of the same circle, they have the same length. We want to know the relationship between the lengths of chords ๐ด๐ต and ๐ท๐ธ if we know that ๐ท๐ธ is farther from the center than ๐ด๐ต. In other words, we assume ๐๐ถ๐๐ถ leads to ๐๐นโ๐๐ถ>0๏จ๏จ, so the left-hand side of this equation must be positive. This means ๐ด๐ถโ๐ท๐น>0,๐ด๐ถ>๐ท๐น.๏จ๏จ๏จ๏จwhichleadstoSince ๐ด๐ถ and ๐ท๐น are positive lengths, we can take the square root of both sides of the inequality to obtain ๐ด๐ถ>๐ท๐น. This leads to the following Relationship between the Lengths of Chords and Their Distance from the CenterConsider two chords in the same circle whose distances from the center are different. The chord that is closer to the center of the circle has a greater length than the theorem allows us to compare the lengths of chords in the same circle based on their distance from the center of the circle. In our first example, we will apply this theorem to obtain an inequality involving 1 Comparing Chord Lengthes based on their Distances from the CenterSupposed that ๐ต๐ถ=8cm and ๐ต๐ด=7cm. Which of the following is true?๐ท๐=๐๐๐ท๐>๐๐๐ท๐๐ต๐ด, which means that chord ๐๐ is closer to the center. Hence, the length of chord ๐๐ is greater than that of the other true option is C, which states that ๐ท๐๐๐ธ, find the range of values of ๐ฅ that satisfy the data We recall that for two chords in the same circle, the chord that is closer to the center of the circle has a greater length than the other. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the this example, we have two chords, ๐ด๐ต and ๐ถ๐ท. Since ๐๐ธ intersects perpendicularly with chord ๐ด๐ต, length ๐๐ธ is the distance of this chord from the center. Similarly, length ๐๐น is the distance of chord ๐ถ๐ท from the center. Since we are given ๐๐น>๐๐ธ, we know that chord ๐ด๐ต is closer to the center. This leads to the fact that chord ๐ด๐ต has a greater length than chord the given diagram, we note that ๐ด๐ต=๐ฅ+4cm and ๐ถ๐ท=24cm. Hence, the inequality ๐ด๐ต>๐ถ๐ท can be written as ๐ฅ+4>24,๐ฅ> this only provides the lower bound for ๐ฅ. To identify the upper bound for ๐ฅ, we should ask what the maximum length of chord ๐ด๐ต is. Since the length of a chord is larger when it is closer to the center, the longest chord should occur when the distance from the center is zero. If the distance of a chord from the center is zero, the chord should contain the center. In this case, the chord is a diameter of the circle. Since the radius of the circle is 33 cm, its diameter is 2ร33=66cm. This tells us that the length of ๐ด๐ต cannot exceed 66 cm. Additionally, since ๐ด๐ต in the given diagram does not contain the center ๐, we know that the length of chord ๐ด๐ต must be strictly less than 66 cm. Hence, ๐ฅ+4<66,๐ฅ< gives us the upper bound for ๐ฅ. Combining both lower and upper bounds, we have 20<๐ฅ< interval notation, this is written as ]20,62[.In previous examples, we considered the relationship between the lengths of two chords in the same circle and their distances from the center of the circle when the distances are not the equal. Recall that two circles are congruent to each other if the measures of their radii are equal. Since the proof of this relationship only uses the fact that the radii of the circle have equal lengths, this relationship can extend to two chords from two congruent can we say about the lengths of chords in the same circle, or in congruent circles, if their distances from the respective centers are equal? It is not difficult to modify the previous discussion to fit this particular case. Consider the following assume that chords ๐ด๐ต and ๐ท๐ธ are equidistant from the center, which means ๐๐ถ=๐๐น. We also know that the radii are of the same length, thus ๐๐ด=๐๐ท. This tells us that the hypotenuse and one other side of the two right triangles โณ๐๐ถ๐ด and โณ๐๐น๐ท are equal. Since the lengths of the remaining sides can be obtained using the Pythagorean theorem, the lengths of the third sides, ๐ด๐ถ and ๐ท๐น, must also be equal. Since these lengths are half of those of the chords, the two chords must have equal lengths. This result can be summarized as Equidistant Chords in Congruent CirclesConsider two chords in the same circle, or in congruent circles. If they are equidistant from the center of the circle, or from the respective centers of the circles, then their lengths are the next example, we will use this relationship to find a missing length of a chord in a given 3 Finding a Missing Length Using Equidistant Chords from the Center of a CircleGiven that ๐๐ถ=๐๐น=3cm, ๐ด๐ถ=4cm, ๐๐ถโ๐ด๐ต, and ๐๐นโ๐ท๐ธ, find the length of We recall that two chords in the same circle that are equidistant from the center of the circle have equal lengths. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the this example, we have two chords, ๐ด๐ต and ๐ท๐ธ. Since ๐๐ถ intersects perpendicularly with chord ๐ด๐ต, length ๐๐ถ is the distance of this chord from the center. Similarly, length ๐๐น is the distance of chord ๐ท๐ธ from the center. From the given information, we note that ๐๐ถ=๐๐น, so the two chords are equidistant from the center of the circle. Hence, the two chords must have equal lengths, ๐ท๐ธ= the diagram above, we are given that ๐ด๐ถ=4. We recall that the perpendicular bisector of a chord passes through the center of the circle. Since ๐๐ถ is perpendicular to chord ๐ด๐ต and passes through center ๐ of the circle, it must be the perpendicular bisector of chord ๐ด๐ต. In particular, this means that ๐ถ is the midpoint of ๐ด๐ต, which gives us ๐ด๐ถ=๐ต๐ถ. Since ๐ด๐ถ=4cm, we also know that ๐ต๐ถ=4cm. Hence, ๐ด๐ต=๐ด๐ถ+๐ต๐ถ=4+4= tells us that the length of ๐ด๐ต is 8 cm. Since we know ๐ท๐ธ=๐ด๐ต, we conclude that the length of ๐ท๐ธ is 8 far, we have discussed implications for the lengths of chords depending on their distance from the center of the circle. We now turn our attention to the converse relationship. More specifically, if we know that two chords in two congruent circles have equal lengths, what can we say about the distance of the chords from the respective centers of the circles? Let us consider the following can label the midpoints of both chords, which are where the blue lines intersect with the chords perpendicularly. Also, we add radii ๐๐ด and ๐๐ท to the diagram. Since the circles are congruent, we know that the lengths of the radii are equal, which leads to ๐๐ด=๐๐ท as seen in the diagram know that ๐ธ and ๐น are midpoints of the chords so ๐ด๐ธ=12๐ด๐ต๐ท๐น= we are assuming that the chords have equal lengths, we know that ๐ด๐ธ=๐ท๐น as marked in the diagram above. This tells us that the hypotenuse and one other side of the two right triangles โณ๐๐ธ๐ด and โณ๐๐น๐ท are equal. Since the lengths of the remaining sides can be obtained using the Pythagorean theorem, the lengths of the third sides must also be equal. This tells us ๐๐ธ= other words, the distances of the chords from the respective centers are equal. We can summarize this result as Chords of Equal Lengths in Congruent CirclesTwo chords of equal lengths in the same circle, or in congruent circles, are equidistant from the center of the circle, or the respective centers of the us consider an example where we need to use this statement together with other properties of the chords of a circle to find a missing 4 Finding a Missing Length Using Equal ChordsGiven that ๐ด๐ต=๐ถ๐ท, ๐๐ถ=10cm, and ๐ท๐น=8cm, find the length of We recall that two chords of equal lengths in the same circle are equidistant from the center of the circle. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the this example, we have two chords, ๐ด๐ต and ๐ถ๐ท. Since ๐๐ธ intersects perpendicularly with chord ๐ด๐ต, length ๐๐ธ is the distance of this chord from the center. Similarly, the length ๐๐น is the distance of chord ๐ถ๐ท from the center. Since we are given ๐ด๐ต=๐ถ๐ท, we know that the chords have equal lengths. This leads to the fact that the chords are equidistant from the center ๐๐ธ= we are looking for length ๐๐ธ, it suffices to find length ๐๐น instead. We note that ๐๐น is a side of the right triangle โณ๐๐ถ๐น, whose hypotenuse is given by ๐๐ถ=10cm. If we can find the length of side ๐ถ๐น, then we can apply the Pythagorean theorem to find the length of the third side, find length ๐ถ๐น, we recall that the perpendicular bisector of a chord goes through the center of the circle. Since ๐๐น perpendicularly intersects chord ๐ถ๐ท and goes through center ๐, it is the perpendicular bisector of the chord. Hence, ๐ถ๐น=๐ท๐น. Since ๐ท๐น=8cm, we obtain ๐ถ๐น= the Pythagorean theorem to โณ๐๐ถ๐น, ๐๐น+๐ถ๐น=๐๐ถ.๏จ๏จ๏จSubstituting ๐๐ถ=10cm and ๐ถ๐น=8cm into this equation, ๐๐น+8=10,๐๐น=100โ64=36.๏จ๏จ๏จ๏จwhichleadstoSince ๐๐น is a positive length, we can take the square root to obtain ๐๐น=โ36= that since ๐๐ธ=๐๐น, we conclude that the length of ๐๐ธ is 6 our final example, we will use the relationship between lengths of chords and their distances from the center of the circle to identify a missing 5 Finding the Measure of an Angle in a Triangle inside a Circle Where Two of Its Vertices Intersect with Chords and Its Third Is the Circleโs CenterFind ๐โ We recall that two chords of equal lengths in the same circle are equidistant from the center of the circle. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the this example, we have two chords ๐ด๐ต and ๐ด๐ถ that have equal lengths. We recall that the perpendicular bisector of a chord goes through the center of the circle. Since ๐ and ๐ are midpoints of the two chords and ๐ is the center of the circle, line segments ๐๐ and ๐๐ must be the perpendicular bisectors of the two chords. In particular, these lines intersect perpendicularly with the respective chords. This tells us that ๐๐ and ๐๐ are the respective distances of chords ๐ด๐ต and ๐ด๐ถ from the center of the the two chords have equal lengths, they must be equidistant from the center. This tells us ๐๐= also tells us that two sides of triangle ๐๐๐ have equal lengths. In other words, โณ๐๐๐ is an isosceles triangle. Hence, ๐โ ๐๐๐=๐โ also know that the sum of the interior angles of a triangle is equal to 180โ. We can write ๐โ ๐๐๐+๐โ ๐๐๐+๐โ ๐๐๐=180.โWe know that ๐โ ๐๐๐=102โ and also ๐โ ๐๐๐=๐โ ๐๐๐. Substituting these expressions into the equation above, 102+2๐โ ๐๐๐=180,2๐โ ๐๐๐=180โ102=78.โโwhichleadstoTherefore, ๐โ ๐๐๐=782=39โ.Let us finish by recapping a few important concepts from this PointsThe distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the two chords in the same circle, or in two congruent circles, whose distances from the center, or the respective centers, are different. The chord that is closer to the respective center is of greater length than the two chords in the same circle, or in congruent circles. If they are equidistant from the center of the circle, or from the respective centers of the circles, their lengths are chords of equal lengths in the same circle, or in congruent circles, are equidistant from the center of the circle, or the respective centers of the circles.iwnks. 427 194 446 356 15 104 488 321 1